Information vs Dimension -- an Algorithmic Perspective

This paper surveys work on the relation between fractal dimensions and algorithmic information theory over the past thirty years. It covers the basic development of prefix-free Kolmogorov complexity from an information theoretic point of view, before introducing Hausdorff measures and dimension alon...

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Bibliographic Details
Published inarXiv.org
Main Author Reimann, Jan
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.08.2024
Subjects
Complexity
Entropy (Information theory)
First principles
Fractal geometry
Information theory
Mathematics - Logic
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Summary:This paper surveys work on the relation between fractal dimensions and algorithmic information theory over the past thirty years. It covers the basic development of prefix-free Kolmogorov complexity from an information theoretic point of view, before introducing Hausdorff measures and dimension along with some important examples. The main goal of the paper is to motivate and develop the informal identity "entropy = complexity = dimension" from first principles. The last section of the paper presents some new observations on multifractal measures from an algorithmic viewpoint.
ISSN:2331-8422
DOI:10.48550/arxiv.2408.05121